Make softbodys squeeze through other objects?
Consider Using a Lattice Modifier which can squeeze or inflate 3D meshes.  



If you are not being extremely particular this is an easier choice.  Since the 3D object in the video might be a simple sphere this may simplify your work.  You will still have work to do as you may be concerned with more degrees of realism regarding the volume compression and expansion.   The lattice has shape keys.  The lattice and the sphere can be moved relative to each other for animation.  I am letting you be the artist here if you like.  You can definitely improve the lattice shape with experimentation.

If wobble suits your tastes then you can add that with another lattice.  I could try to convince you that the video included your video has low realism wobble.  You can view and be the judge of that in the video.

There are also more difficult and complicated sequences of modifiers such as weight modifiers and shrink wraps.